{
  "id": "1112.6230",
  "version": "v3",
  "published": "2011-12-29T05:20:56.000Z",
  "updated": "2015-03-04T02:19:05.000Z",
  "title": "Jet schemes and invariant theory",
  "authors": [
    "Andrew R. Linshaw",
    "Gerald W. Schwarz",
    "Bailin Song"
  ],
  "comment": "Final version, to appear in Annales de l'Institut Fourier",
  "categories": [
    "math.AG",
    "math.GR",
    "math.RT"
  ],
  "abstract": "Given a complex, reductive algebraic group $G$ and a $G$-module $V$, the $m$th jet scheme $G_m$ acts on the $m$th jet scheme $V_m$ for all $m\\geq 0$. We are interested in the invariant ring $\\mathcal{O}(V_m)^{G_m}$ and whether the map $p_m^*\\colon\\mathcal{O}((V//G)_m) \\rightarrow \\mathcal{O}(V_m)^{G_m}$ induced by the categorical quotient map $p\\colon V\\rightarrow V//G$ is an isomorphism, surjective, or neither. Using Luna's slice theorem, we give criteria for $p_m^*$ to be an isomorphism for all $m$, and we prove this when $G=SL_n$, $GL_n$, $SO_n$, or $Sp_{2n}$ and $V$ is a sum of copies of the standard representation and its dual, such that $V//G$ is smooth or a complete intersection. We classify all representations of $\\mathbb{C}^*$ for which $p^*_{\\infty}$ is surjective or an isomorphism. Finally, we give examples where $p^*_m$ is surjective for $m=\\infty$ but not for finite $m$, and where it is surjective but not injective.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-08-12T18:33:36.000Z",
      "abstract": "Given a complex, reductive algebraic group G and a G-module V, the mth jet scheme G_m acts on the mth jet scheme V_m for all m\\geq 0. We are interested in the invariant ring O(V_m)^{G_m} and whether the map p_m^*: O((V//G)_m) \\rightarrow O(V_m)^{G_m} induced by the categorical quotient map p: V \\rightarrow V//G is an isomorphism, surjective, or neither. Using Luna's slice theorem, we give criteria for p_m^* to be an isomorphism for all m, and we prove this when G=SL_n, GL_n, SO_n, or Sp_{2n} and V is a sum of copies of the standard representation and its dual, such that V//G is smooth or a complete intersection. We classify all representations of \\mathbb{C}^* for which p^*_{\\infty} is surjective or an isomorphism. Finally, we give examples where p^*_m is surjective for m=\\infty but not for finite m, and where it is surjective but not injective.",
      "comment": "Major revision. A significant error has been corrected, several new results added",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-03-04T02:19:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "invariant theory",
      "mth jet scheme",
      "lunas slice theorem",
      "isomorphism",
      "complete intersection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.6230L"
    }
  }
}